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Sta 2191 Financial Mathematics I Question Paper
Sta 2191 Financial Mathematics I
Course:Bachelor Of Science In Actuarial Science
Institution: Dedan Kimathi University Of Technology question papers
Exam Year:2011
KIMATHI UNIVERSITY COLLEGE OF TECHNOLOGY
University Examinations 2011/2012
FIRST YEAR SECOND SEMESTER EXAMINATION FOR THE DEGREE OF
BACHELOR OF SCIENCE IN ACTUARIAL SCIENCE
STA 2191 Financial Mathematics I
DATE: 6TH DECEMBER 2011 TIME: 11.00 AM - 1.00 PM
Instructions: Answer QUESTION ONE and any other TWO QUESTIONS.
QUESTION ONE (30 marks) (COMPULSORY)
(a). De ne the following investments and describe the associated cash
ows:
(i). Index linked security
(ii). Call deposits
(iii). Equity [6 marks]
(b). Calculate the accumulated value after 5 years of an investment of 1,500 made now for
the following interest rate scenarios:
(i). At simple rate of interest of 6% p.a. for rst 3 years, reinvested at a simple rate of
interest of 7% p.a. for the next 2 years and 3 months. [2 marks]
(ii). At interest rate of 5% p.a. convertible monthly for rst 21
2 years; 2% per quarter
for next 1 year; and at a force of interest of 8% p.a. thereafter. [3 marks]
(c). The rate of discount per annum convertible quarterly is 8%. Calculate up to 4 decimal
places:
(i). The equivalent rate of interest per annum convertible half-yearly [2 marks]
(ii). The equivalent rate of discount per annum convertible monthly [1 marks]
(iii). The equivalent rate of interest per annum convertible once in two years. [2 marks]
(d). An investor deposits 25,000, then withdraws level annual amounts starting two years
after the deposit was made. If immediately after the 10th annual drawing, the investor
has 1,600 left in the account, nd the amount of each withdrawal if the annual rate of
interest is 8%. [4 marks]
(e). Calculate s(4)
2:5
at an e ective rate of interest of 8% p.a. [4 marks]
(e). Fund A accumulates at 9% e ective and Fund B at 8% e ective. At the end of 10 years,
the total of the two funds is 52,000. At the end of 8 years, the amount in Fund B is three
times that in Fund A. How much is in Fund A after 15 years? [6 marks]
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QUESTION TWO (20 marks) (Optional)
(a). A 3-year annuity has the following payment schedule:
Year 1 6,500 per annum paid continuously
Year 2 6,500 per annum paid monthly in advance
Year 3 6,500 per annum paid half yearly in advance.
Calculate the total value of these payments at the beginning of the rst year at a rate of
interest of 9% per annum convertible quarterly. [8 marks]
(b). A single investment of $500 is accumulated at a nominal rate of discount of 6% p.a.
convertible half-yearly for 1 year, followed by a nominal rate of interest of 6% p.a. convertible
every 4 months for 1 year. Calculate the accumulated amount of this investment
after 2 years. [4 marks]
(c). (i). Derive the annuity function (Ia)n . [4 marks]
(ii). Find the present value at 9% e ective of a 20-year annuity, with the rst payment
due immediately, in which the payments follow the pattern
1; 4; 9; 16; ; 400
. [4 marks]
QUESTION THREE (20 marks) (Optional)
(a). The force of interest, (t), is a function of time and at any time t, measured in years, is
a+bt where a and b are constants. An amount of 4,500 invested at time t = 0 accumulates
to 5,500 at time t = 5 and 12,000 at time t = 10.
(i). State the principle of consistency and show that it holds. [5 marks]
(ii). Calculate the values of a and b. [6 marks]
(iii). Calculate the constant force of interest per annum that would give rise to the same
accumulation from time t = 0 to time t = 10. [2 marks]
(b). Given that i = 0:15 nd the present value of an annuity of 100 per year continuing forever
if
(i). the rst payment is due in one year
(ii). the rst payment is due immediately
(iii). the rst payment is due in 5 years. [7 marks]
QUESTION FOUR (20 marks) (Optional)
(a). Kamau takes a loan which is to be repaid in instalments annually in arrears. The rst
instalment is 160, the second 155 and so on with the repayments reducing by 5 p.a. until
the end of the 15th year after which there are no further payments.
The rate of interest charged by the lender is 8% p.a. e ective.
(i). Calculate the amount of the loan. [4 marks]
(ii). Calculate the interest and capital components of the third payment. [4 marks]
(iii). Calculate the amount of capital repaid in the instalment at the end of the thirteenth
year. [4 marks]
(a). Derive the annuity function (Ia)n . If (t) = 0:01t for 0 < t < 5 , calculate the value of
(Ia)5 . [8 marks]
2
QUESTION FIVE (20 marks) (Optional)
(a). (i). Prove that
1
d(m)
??
1
i(m)
=
1
m
. [4 marks]
(ii). Evaluate the following annuity functions at an e ective rate of interest i = 16%
a(4)
5
and s(12)
5:5
[5 marks]
(b). A vendor has two o ers for a house: (i) 40,000 now and 40,000 two years hence, or (ii)
28750 now, 23750 in one year, and 27,500 two years hence. He makes the remark that
one o er is \just as good" as the other. Find the two possible rates of interest which
would make his remarks correct. [6 marks]
(c). The force of interest (t) is a function of time and at any time, measured in years, is
given by the formula:
(t) =
8>><
>>:
0:04 + 0:01t 0 t 4
0:12 ?? 0:01t 4 < t 8
0:06 8 < t
Calculate the present value at time t = 0 of a payment stream, paid continuously from
time t = 9 to t = 12, under which the rate of payment at time t is 50e0:01t. [5 marks]
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